Abstract and Applied Analysis

Volume 2013 (2013), Article ID 725495, 12 pages

http://dx.doi.org/10.1155/2013/725495

Research Article

Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion

Xiaojing Yu, Peixuan Weng, and Yehui Huang

School of Mathematics, South China Normal University, Guangzhou 510631, China

Received 17 November 2012; Revised 11 March 2013; Accepted 21 March 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Xiaojing Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study a competing pioneer-climax species model with nonlocal diffusion. By constructing a pair of upper-lower solutions and using the iterative technique, we establish the existence of traveling wavefronts connecting the pioneer-existence equilibrium and the coexistence equilibrium. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts as .

1. Introduction

As we know, the interactions among species are important in determining the process of evolution for the ecosystem, and the modeling accompanied with the mathematical analysis of the models can help people to understand and control the propagation of species. In general, the per capital growth rate (i.e., fitness) for a species in the model is assumed to be a function of a weighted total density of all interacting species. A well-known example is the standard Lotka-Volterra model; its fitness of a species is a linear function. It is natural to consider other kinds of fitness functions other than the linear one, because of the various species and interaction rules. In this paper, we will analyze a reaction-diffusion model describing pioneer and climax species. This model describes interaction among species with peculiar fitness functions.

A species is called a pioneer species if it thrives best at lower density but its fitness decreases monotonically with total population density for overcrowded. Thus, the fitness function of a pioneer species is assumed to be a decreasing function. Pine and yellow poplar are the species of this type. A species is called a climax species if its fitness increases up to a maximum value and then decreases of its total density. Hence, a climax population is assumed to have a nonmonotone, “one-humped” smooth fitness function. Oak and maple are the climax species.

A typical reaction-diffusion model for a pioneer-climax species is given by the following system:

where and represent densities of the pioneer and climax species, respectively. and denote the pioneer fitness function and climax fitness function, respectively, . By making changes of variables , , system (1) changes into the form (the tildes of and are dropped out)

where we still use and as the new coefficients without confusing.

From the previous introduction, we assume that the pioneer fitness function satisfies

for some , and the climax fitness function satisfies

Ricker [1] used the fitness function , Hassell and Comins [2] used the fitness function , and Cushing [3] used the fitness function . It is obvious that these and have the curves in Figure 1.

(a)

(b)

Figure 1: Typical fitness functions for pioneer species and climax species.

There are some existing results about the stability and traveling wave solutions for (2) ([4–6]). About traveling wave solutions, Brown et al. [4] studied the traveling wave of (1) connecting two boundary equilibria by singular perturbation technique, and Yuan and Zou [6] obtained the existence of traveling wave solutions connecting a monoculture state and a coexistence state by upper-lower solution method combined with the Schauder fixed point theorem. Also see van Vuuren [7] for the existence of traveling plane waves in a general class of competition-diffusion systems and Murray [8] for more biological description of traveling wave solutions.

For system without spatial diffusion, the model will be

Selgrade and Roberds [9], Sumner [10] analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong [11], Sumner [12] considered the stable periodic behavior of (5). Because of the existence of rich equilibria and the various ranges of parameters, the dynamics of ordinary differential system (5) are complex, and a detailed review of all equilibrium types can be found in Buchanan [13, 14].

Although the Laplacian operator is always used to model the diffusion of the species, it suggests that the population at the location can only be influenced by the variation of the population near . As we know that the individuals can move freely, then the movement of individuals is bounded to affect the other individuals. So, the Laplacian operator may have some shortage to describe the diffusion. One way to deal with this problem is to replace the Laplacian operator with a convolution diffusion term This implies that the probability distribution function for the population at location moving to the location is . At time , the total individuals that move from the whole space into the location will be . Therefore, one may call it as a nonlocal diffusion, and, correspondingly, call as a local diffusion. During the recent years, the models with the nonlocal diffusion have been attracted much more attentions (see [15–18]).

In this paper, instead of (2), we will concentrate on the following pioneer-climax species with nonlocal diffusion:

where , are positive constants accounting for the diffusivity, is a kernel function which is continuous on satisfying

We are interested in traveling wavefronts accounting for a mildinvasion of the two species (traveling wavefronts connecting a boundary equilibrium and the coexistence equilibrium). For system (2), the sufficient condition for (2) to have a mildinvasion is (see [6]), but our condition in this paper for system (6) is , which reveals a fact that the nonlocal diffusion of either the pioneer species or the climax species did affect the climax invasion and wave propagation. Please see Section 5 for the discussion.

The remaining of this paper is organized as follows. In Section 2, there are some preliminaries about the equilibria and the system is transformed into a cooperative one. In Section 3, we prove the existence of traveling wavefronts by using an iteration scheme combined with a pair of admissible upper and lower solutions, which can be constructed obviously, and thus a criterion of the existence for traveling wavefronts is obtained. We also give a discussion on asymptotic behavior for the traveling wavefront tail as in Section 4. At last, we give some concluding discussions in Section 5.

2. Preliminaries

It is evident that is a trivial equilibrium of (6). The system (6) has at least four equilibria and at most six equilibria. The existence of nonnegative steady states depends on the locations of the three nullclines:

The long-term behavior of solutions to (6) can be qualitatively different caused by the different number, distribution, and types of equilibria. The dynamics of the system (6) are of course very rich and complex. However, in this paper, we will only consider the following case:

The condition follows as a sequence. Under the previous assumption, (6) has four nontrivial equilibria: , , , and except for , where

It is obvious that . We further assume that

for the technical reason. See Figure 2 for this situation.

Figure 2: One coexistence state for the two species with .

As mentioned in the introduction, we are interested in the coexistence of the two species. That means that we will seek traveling wavefronts connecting equilibrium and equilibrium .

By making changes of variables , and dropping the tildes, system (6) becomes

and the equilibria ,   are changed into , , respectively, where , .

3. Existence of Traveling Wavefronts

A traveling wavefront of (6) connecting equilibria and can be changed into a traveling wavefront of (11) connecting and . Therefore, we consider the system (11) hereby.

A traveling wave solution of (11) is a solution with the form and , where and is a wave speed. A traveling wavefront is a traveling wave solution which has finite limits . Denoting the traveling wave coordinate still by , we derive the wave profile system from (11):

Associated with (12), we consider its solutions subject to the following boundary value conditions:

For , , implies ; implies but ; implies . Furthermore, the norm in is the Euclidean norm. Define

For some constants , , letting , , we define an operator by

Then, (12) can be written as an equivalent form:

Denote by

It is obvious that a traveling wave solution of the problem (12) and (13) is a fixed point of and vice verse.

The following lemma states the monotone property of .

Lemma 1. Assume that holds, for sufficiently large and , with and ,  ; one has

 (i),     ,  (ii),     

for all .

Proof. In order to prove (i), let , . Then

For , and sufficiently large , it follows that , . Thus, if and for , we have

For (ii), we know that and for . It follows that

From , , we have

Note that for . This leads to

The proof is complete.

The conclusion of the following lemma is direct.

Lemma 2. Assume that , are sufficiently large. For with nondecreasing on , , are also nondecreasing on .

We can easily see that also enjoys the same properties as those for settled in Lemmas 1 and 2.

Let and

It is clear that is a Banach space equipped with the norm defined by .

Definition 3. A pair of continuous functions , is called an upper solution and a lower solution of (12), respectively, if there exists a set such that and are differentiable in and the essential bounded functions satisfy

for .

In what follows, we assume that (12) has an upper solution and a lower solution , such that

(P1) for ;(P2), ;(P3) and are nondecreasing.

Define the following profile set by

It is obvious that is nonempty.

For and , define , and

Lemma 4. For , the functions , and , defined by (26) satisfy

(i); (ii) for ;(iii) and is a pair of upper and lower solutions of (12);(iv) and are continuously differentiable on .

Proof. We only give the argument for , and the situation for can be obtained by mathematical induction. From Definition 3, we obtain

Let , . For any , there exists some such that or . We then derive

where .

By similar arguments, we can get

The previous arguments implies that the conclusion (ii) holds.

From the monotone property of , we can easily obtain that , are nondecreasing for , and therefore the conclusion (i) holds.

For , by Lemma 1, we have

In a similar way, we can prove that

for . This indicates that and are a pair of upper and lower solutions of (12).

The conclusion (iv) is obvious. The proof is complete.

Lemma 5. , and the convergence is uniform with respect to the decay norm .

Proof. We have from Lemma 4 that the following limit exists:

It is easy to know that is a closed and convex set. By the nondecreasing property of , we have . In the following, we prove that the convergence is uniform with respect to the decay norm.

Since

for any , there exists a , such that for all ,

Now, we consider the sequences for . Note is nondecreasing on , and thus

By , there exists a positive constant , such that for . Similarly, we can prove that there exists a positive number , such that for .

From the previous estimates, we know that is equicontinuous on with respect to the supremum norm. On the other hand, we have from Lemma 4 (ii) that is uniformly bounded. By Arzéla-Ascoli theorem, there exist subsequences of which are uniformly convergent in . Without loss of generality, we still express this subsequence as . Thus, there exists a positive integer , such that

Furthermore, we have

Summarizing the previous arguments, for we have

The proof is complete.

Theorem 6. Assume that holds; if (12) has a pair of upper and lower solutions that satisfy (P1)–(P3), then the system (11) has a traveling wavefront satisfying (13).

Proof. By the Lebesgue's dominated convergence theorem and the iteration scheme (26), we have

Therefore, is a fixed point of , which also satisfies (12). Furthermore, (P2) indicates that satisfy . On the other hand, we have from the monotonicity of that exists. Furthermore, since , we know that . By using L’Hôspital’s rule, we obtain

Similarly, one can obtain . That is, is an equilibrium of (12). Note that the assumption (10) implies that there is only one positive equilibrium of (12) satisfying . Therefore, and is a traveling wavefront satisfying (13). The proof is complete.

In order to construct a pair of admissible upper-lower solutions for (12), we linearize (12) at and obtain

Thus, we consider the following characteristic equation:

Note that

The convex property of leads to for any . In view of the previous observation, we have the following lemma directly.

Lemma 7. The following conclusions are true.

(i)There exists a , , such that

(ii)For , for .(iii)For , the equation has two zeros such that

Now we are ready to construct the upper solution of (12).

Lemma 8. Define

Then for , is an upper solution of (12).

Proof. Let be such that

Notice that , we have .

If , , , we have from the fact for and that

Similarly, we have from the fact for , and for that

If , , , we have

The proof is complete.

Let satisfying , we can obtain from (45) that

Lemma 9. Define

where is a constant to be chosen later. Then is a lower solution of (12). Furthermore,

is a lower solution of (12) satisfying

Proof. Let be such that , it follows that

If , , , and for , we have

Let By the fact that

we have

Note that (58) leads to for ; it follows that

and thus

Therefore,

Let sufficiently large; we can have

hence,

If , , , and for , we have

From the previous arguments, we obtain that is a lower solution of (12). By Lemma 4, we can get that is also a lower solution. Furthermore, for , , by direct calculation, we have for . Therefore, for . Similarly, we have for . The proof is complete.

Theorem 10. Assume that and hold. Then for any , the system (11) has a traveling wavefront with speed , which connects and .

Proof. The conclusion for can be obtained from the previous discussions. We only need to establish the existence of wave fronts when .

Let with . For , (12) with admits a nondecreasing solution such that

Without loss of generality, we assume that . Obviously, , , and satisfy

As the same argument in Lemma 5, we can obtain that is uniformly bounded and equicontinuous on ; using Arzéla-Ascoli theorem and the standard diagonal method, we can obtain a subsequence of , still denoted by , such that uniformly for in any bounded subset of , as . Clearly, is nondecreasing and .

By the dominated convergence theorem and (67), it follows that

Since and exist, using L’Hôspital rule leads to

Thus, is a traveling wavefront of the system (11) connecting and .

Remark 11. We say that the is the minimal wave speed in the sense that (11) has no traveling wavefront with . We could briefly explain this in the following. In fact, the linearization of (12) at zero solution is (41), and the function is obtained by substituting in the second equation of (41). For , we know from (ii) of Lemma 7 that for any . We have from the second equation of (12) and the second equation of (41) that (12) cannot have a solution that satisfies .

Theorem 12. Assume that and hold. Then for any , the system (6) has a traveling wavefront with speed c, which connects and .

4. Asymptotic Behavior for Traveling Wavefronts

In this section, we discuss the asymptotic behavior for the traveling wavefronts obtained in the previous section as .

Theorem 13. Let be a traveling wavefront of (11) decided by Theorem 10; then

where is the smallest root of the characteristic equation (42).

Proof. Note that

Then we have

which implies that

Note that we have from and that

which is uniformly on . By the second equation of (12), we have from (73), and convergence theorem that

Since , by (17), we know that

On the other hand, by (76), (15), (42), and (75), noting and using L’Hôspital’s rule, we get

By the first equation of (12), we have from (75) and that

Combining (77) and (78), and noting that , we obtain from a direct calculation that

Now, letting be a traveling wavefront of (6) decided by Theorem 12, by the equivalence between (6) and (11), we know that and thus obtain the following theorem.

Theorem 14. Let be a traveling wavefront of (6) decided by Theorem 12; then

where is the smallest root of the characteristic equation (42).

5. Concluding Discussions

We have considered a reaction model with nonlocal diffusion for competing pioneer-climax species. Some recent works (see Brown et al. [4], Yuan and Zou [6]) showed that the model with local diffusion (expressed by Laplacian operator) supports traveling wavefronts connecting two boundary equilibria or one boundary equilibrium and the coexisting equilibrium under some restrictions on the parameters. In Yuan and Zou [6], the sufficient condition for (2) to have the traveling wavefront connecting and is , but ours for (6) with nonlocal diffusion is . For a fixed , to shift to , one needs a smaller . But for a fixed , to shift to , one needs a larger . These facts imply that the nonlocal diffusion of the pioneer species accelerates the mild wave propagation, while the nonlocal diffusion of the climax species defers the mild wave propagation. That is to say, the nonlocal diffusion did affect the wave propagation of these two competitive species.

The generalized boundary conditions,

lead to an explanation that the climax species starts its invasion after that the pioneer species has achieved its steady state, and also the competition between the two species was not intense; they can achieve a coexistence state finally. See [19] for the significance of biological invasion.

We believe that this is the first time that the dynamics of the pioneer-climax competition model with nonlocal diffusion are studied. This model has complicated equilibrium structure, and we only considered one possible case about the traveling wavefront connecting one boundary equilibrium and the coexisting equilibrium. Some other situations about the species invasions and propagation of waves would be of great interest for further research.

Acknowledgments

The authors are very grateful to the anonymous referees for careful reading and helpful suggestions which led to an improvement of their original paper. Research is supported by the Natural Science Foundation of China (11171120), the Doctoral Program of Higher Education of China (20094407110001), and the Natural Science Foundation of Guangdong Province (10151063101000003).

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